TAYLOR'S THEOREM. 



T.YYl.olfS THEOREM. 



tenet has been used, as far at three terms, in the article 



All that precedes U found In elementary treatises, with the exception 

 of a few terms of the last series : we now come to matter which has 

 been hitherto only the property of the well-read mathematician, but 

 which wall deserve! to be made as common as Taylor's Theorem. We 

 refer to Arbogast's swtAorf of dm ratio**. [ARBOOAST, in Bioo. Div.] 

 Few, even among mathematicians, are aware of the power of this 

 process, which may perhaps arise from their taking Lacroix a account 

 of it, instead of consulting the work of Arbogast himself : the former 

 hat only exhibited it to show that it may be reduced to processes of 

 the differential calculus ; and even the latter has so loaded his method 

 with heavy applications, that he has concealed much of itt beauty and 



The foundation of Arbogast's methods is a contrivance for expediting 

 the expansion of f (a + 6x + tr 5 + ....) into a series of the form A + B.r 



+ or* ) The process by which B is formed from A, c from B, 



Ac. it uniform, and is called derivation ; and A being fa, B may be 

 called Ufa, c may be called DDfa, or irfa, and to on. Hence 4 ought 

 to be called Da, c ought to be 0*a, and so on. This notation is not 

 precisely that of Arbogast, but will do for our purpose. For more 

 detail,* tee the Differential Calculus (' Library of Useful Knowledge '), 

 pp. 838-334. 



If, for a moment, we write the expansion thus 



f (a,, -f a,.r + a,^* + Ac.) =A -f A,I + A,* 8 *- Ac. 



and if we differentiate both sides with respect to a m , x and all the 

 other coefficients remaining constant, we have 



which shows that a, cannot enter any coefficient preceding A., or 



The first side of this is the same series, whatever letter a. was made 

 to vary ; the second side is therefore always the same series ; whence 

 we collect that d\ m v . : rfo. does not alter with the value of m, being 

 always the coefficient of x* in the development of f '(a,, + a,* * , Ac.). 

 It U enough to satisfy this condition for each letter and its preceding 

 one ; that is to say, each coefficient differentiated with ree]iect to any 

 one letter, is to yield the same result as the directly preceding co- 

 efficient differentiated with respect to the directly preceding letter. 

 The following rules arc found sufficient. To past from any one de- 

 rivative of fa to the next, arrange the letters a, 4, c, Ac., or a ol a,, a.,, 

 Ac., whichever may be used, in order, in every term : differentiate with 

 respect to the last letter in each term, and multiply by the letter 

 which comes next to it. And when the last but one immediately 

 precedes the hut in the alphabet or other consecutive system, do the 

 same with the last but one, and divide by the exponent of the last 

 letter, as it becomes after the increase which it receives from the 

 process of the preceding letter; but in no case use any letters but the 

 last or the hut but one. For instance, beginning with fa, in which is 

 only one letter, we have f 'a . 6, or 



Dfa=f'a. A; 



'n which are two letters, a and 6, consecutive. Operate upon 6, and 

 we have f 'a . c ; operate on f 'a, and we have again f "a . 4, which, with 

 the 4 which was in before, is f "a . b 1 , which we divide by the new 

 exponent of 6, or by 2, whence 



f"a 

 D'fa = f a . c + * H. 



In forming rPfa, we use only c in fa. c, because a does not immc- 

 dtetdh 

 *,Ac.j 



ly precede c ', and we get (the succession being a, b, e, e,f, <j, h, 

 ) 



Dfa = f a . e + -jr . 26c + ^ If; 



and to on. At toon however as the law is established, it is best to 

 form a table of the successive derivatives of the powers of 4 by this 

 tame liw : we then have 



D* fa = f 'aD"-'4 + - D"-' 4 s + - 1 D"- 3 V + Ac. 



as far as r , 6* ; 



in which fa, fa, Ac. are to be taken Srom the function by common 

 differentiation, and the derivatives of the power* of 6 from the t .I.;.-. 

 This being done, we have 



f(a -f 6x + c* + at* + fa* + y* + , Ac.) = 

 fa + Dfa . x + D'fa . x* + D'fa . x* + , Ac. 



and the process it shortened to itt utmost extent; all that is not 



Tber to s treat deal on the subject in the ' Mathematical Trraliiea ' (pool- 

 honxm.) of the Rer. John Went, publiubcd at Edinburgh In 1838. Mr. Wc.t 

 h. raUtltuted a notation, for that of Arbogait, in which ho will probably hare 

 few follower!. The ntudent who to not repelled by tbii, and cannot procure 

 "i work, will And Wctfi treatises abounding in derivations. 



differentiation being manly reference to a table and writing the 

 rttoll 



We shall give materials for proceeding as far at the term D 1! fa . .T", 

 not that so much will often be necessary, but because it is desirable t.> 

 show with how little trouble questions of enormous labour in tin- 

 ordinary way, such, for instance, as that solved in REVKRSION or 

 SKRIF.S, may be looked at without dismay. We have to form every 

 derivative of every power of o,D'i*, in which m + N doe* not ex- 

 ceed 12. 



DO=C 

 Do=A 

 Di= 



D*6=i- 



vb-l 



n'4 = o 



Dt'=2ie 

 Dft = 2of 

 DV=26/-t-2 



D'A 8 -= 26m + 2cl + 2dc + -J fit + <f 

 D'i = 24 + 2cm + 2el t Wt + ^ . /, 



2*p + 2cn + 2em + 2JI + Zyk + If 



= 36>A + 6% + 6te/+ 3c : /+ 3' 

 Dft = 36i + 66cA + 6beg + 3A/ -I- 3c>^ + 60/4- e* 

 D'6 = 84 1 / + Obft + 6br h + 6bfg + 3c*A + 6 cty + 3c/ + Se / 

 D>b*= 3b*m + 6bd + bdc + ebfh, + 36jr ! -r 3<~l- + Cceh + 6rf.j + 8<V 



+ S<!/ 



n'f = 3W + 66cm + 6bel + Sbfk + &*/h + 3c'/ + Oral- -f G>/& + 3cr/ 3 



= 46^- + 126c/+ 6*** + 1 24c'< + c 4 



126>e/+ 126<r'/+ 121ce + t<?e 



4C 5 / 



124e'/+ teg + 12 



+ 124'rit -r 

 1268V + 



244cf/i 



D>4 s =54 4 /t204'-H04 ; c : > - 



D'4 = 5b'g + 204 S <:/+ 1 04V+ 304V + 54c 



D S 4 = 56'A + 204 1 e + 204V 1 306V/-1- 306'cf 4- 204e=e + c 5 



D6* = 54 4 it + 20A 3 cA -t- 206^5 + 104 s / 3 + 



= 56V + 204>cfc + 204>A + ZOVfg + 804 I c'A + QWcrg + 

 + SOtftf/^ 204cV + 604c 2 f/-f 204ce + Sc 4 



D4=66'c 

 D4 i =64+154c 5 

 D a 4*=66/+ 306ce -h 206V 

 D4= 64'o + 304*c/+ 1 56'e 5 (- 604 3 c 3 e + 154V 

 D6= 66 5 A + 306co + 306f/+ 604'c s /+ 606 5 cc= + 606=A + OAc* 

 B^i' = 64** + 304cA + 304Vff + 156'/ + COA 3 ^ + 1 20A J /+ 206V 

 r 604cy + 904'cV -r 304c'e + c 



l)4 7 =74 8 c 

 D4'=76* + - 

 uV = 76y+ 426ce + 856V 

 D'A? = 76j + 42Atf + 21AV+ lOSA'A -t- 354V 

 D'4?=76'A + 42^ + 426V/ + 1056'c s / <- 

 + 216V 



D6< = 

 D4= 



'/+ 56lfce + 5fi4 V 



<j + 666<:/-(- 28W + lC84c 5 <: + "n'-'c 1 



D4=94V 

 D^^SAV 

 r^A' = 94/+ TOfet + 846V 



D-6=106-f4J6 s c s 



To verify these resulU, observe that if we consider each letter as of 





